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Conditional Probability

Conditional Probability#

Definition#

Use this definition to solve the following problems.

  1. In a highschool, there are 287 students. 92 of the students play football. 50 of them play football and basketball. What is the probability a randomly selected football player also plays basketball?

  2. In a sample of 100 adults, 45 of them owned a dog, 30 of them owned a cat and 18 of them owned a dog and a cat. What percentage of those who have a dog also have a cat?

  3. Suppose 15% of the population of a country have auburn hair and are left-handed, while a total of 25% of the population have auburn hair. What percent of those with auburn hair are left-handed?

  4. You are given the following Venn Diagram,

../../../_images/conditional_probability_venn_diagram.jpg

Use this Venn Diagram to answer the following questions.

  1. Find P(A \mid B)

  2. Find P(B \mid A)

  1. You shuffle a deck of playing cards and deal yourself two cards at random.

    1. What is the probability the first card is a king?

    2. What is the probability both cards are kings?

    3. If the first card is a king, what is the probability the second card is a king?

    4. What is the difference between the meanings of part b and part c?

    5. What is the connection between part a, part b and part c?

    1. What is the probability of not getting a pair of kings?

  1. A box contains three coins: two regular coins and a double-sided coin with heads on both sides. You select a coin at random and flip it.

    1. What is the probability it lands heads up?

    2. If it lands heads up, what is the probability you have selected the double-sided coin?

  2. Whilst paging through a magazine, you notice a interesting survey of randomly selected individuals who were each asked two questions. The first question was “Do you believe the Earth is flat or round?”. The second question was, “Do you think Finland is a real country ?” The results are given in the table below,

Flat

Round

Exists

15

120

Doesn’t

12

35

Use this table to answer the following problems.

  1. Draw a Venn Diagram that represents the results.

  2. How many people were included in this survey?

  3. What is the probability a randomly selected in this survey person believes the Earth is flat?

  4. What is the probability a randomly selected in this survey person believes Finland doesn’t exist?

  5. Of the people who believe the Earth is round, what percentage of them believe Finland exists?

  6. If a randomly selected person believes Finland doesn’t exist, what is the probability he or she believes the Earth is flat?

  7. If a randomly selected person believes the Earth is flat, what is the probability he or she doesn’t believe Finland exists?

  8. Spoiler Alert Why is the answer to part f different from the answer to part g?

Sample Space Reduction#

  1. A bag contains 3 red marbles and 4 blue marbles. Two marbles are drawn at random without replacement. If the first marble drawn is red, what is the probability the second marble is blue?

  2. Suppose two fair dice have been tossed.

    1. What is the probability of the faces sum to 5 if they land on different numbers?

    2. If the total of their top faces is found to be divisible by 5, what is the probability that both of them have landed on five?

  3. Five people in a class of students believe math is awesome, while two people (losers) believe math is a blight upon humanity. Suppose you sample two people from this class without replacement.

    1. If the first person selected believes math is awesome, what is the probability the second person believes math is awesome?

    2. If the first person selected is a loser, what is the probability the second person believes math is awesome?

Warning

Challenge Problem Ahead

  1. Ghahramani, 2005 In a small lake, it is estimated that there are approximately 105 fish, of which 40 are trout and 64 are carp. A fisherman caught eight fish. What is the probability that exactly two of them are trout if we know that at least three of them are not?

Hint

Reduce the sample space and then use Combinatorics.

Bayes’ Laws#

  1. 100 sci-fi fans were polled by the reporters at Imporant News Weekly. 64 of those polled preferred Star Wars to Star Trek, while the rest of them, due to poor life choices, preferred Star Trek to Star Wars. Of the people who preferred Star Wars, 75% of them thought The Empire Strikes Back was the best of the film in the ennealogy. Of the people who preferred Star Trek, only half of them thought The Empire Strike Back was the best film in the series.

    1. What percent of people preferred Star Wars over Star Trek and thought The Empire Strikes Back was the best film in the series?

    2. What percent of people preferred Star Trek over Star Wars and thought The Empire Strikes Back wasn’t the best film in the series?

  2. Missing Cards

Solve these problems about a standard deck of 52 cards.

  1. Suppose one of the cards has been lost. What is the probability that a random card drawn from this deck is a heart?

  2. Suppose two of the cards have been lost. What is the probability that a random card drawn from this deck is a heart?

  1. Ghahramani, 2005 Of patients in a hospital, 20% of those with myocardial infarcation have had strokes and 35% of those without myocardial infarcation have had strokes. If 40% of the patients have had myocardial infarcation, what percent of the patients have had strokes?

  2. Ghahramani, 2005 A factory produces its entire output with three machines. Each machine has an error rate that causes it to produce defective units. Machine I produces 50% of the output and has a 4% error rate. Machine II produces 30% of the output and has a 2% error rate. Machine III produces 20% of the output and has a 4% error rate. What percentage of the total output is defective?

  3. Suppose 1% of the United States Population has a serious life-threatening condition that slowly turns their internal organs into goop and drives them insane before an extremely painful death, known as Sejal’s Disease. A test is developed to help diagnose this awful curse on humanity. The test can determine whether an individual does or does not have Sejal’s Disease. If the person has the disease, it will give a positive result 98% of the time. If the person does not have the disease, it will give a false positive 5% of the time. If you take the test and get a positive result, what is the probability you are afflicted with Sejal’s Disease?

  4. Suppose 80% of seniors, 70% of juniors, 50% of sophomores and 30% of freshman use the Maker Space at Bishop Walsh. If 30% of all students are freshmen, 25% are sophomores, 25% are juniors and 20% are seniors, what percent of all students use the Maker Space?

Independence#

  1. Cards, Single Draw

You are dealt a single card from a standard deck of 52 cards, face down.

  1. Without any prior knowlesdge, what is the probability of the card being a spade?

  2. The dealer reveals to you the face of the card is a Jack. How does this information change your answer to part a?

  3. Based on your answers to the parts a and b, is the event of getting a Jack independent of the event of getting a spade?

2 Balls, With Replacement

Five red balls and two balls are placed into a magical probability box. Two balls are selected with replacement.

  1. Without any prior knowledge, what is the probability the first ball is red?

  2. Without any prior knowledge, what is the probability the second ball is red?

  3. Without any prior knowledge, what is the probability the first ball is blue?

  4. Without any prior knowledge, what is the probability the second ball is blue?

  5. If the first ball is blue, what is the probability the second ball is red?

  6. If the first ball is red, what is the probability the second ball is red?

  7. If the first ball is blue, what is the probability the second ball is blue?

  8. If the first ball is red, what is the probability the second ball is blue?

  9. Based on your answers to part a through part h, is the event of getting a red ball on the first draw independent of the event of getting a blue ball on the second draw? Is the event of getting a blue ball on the first draw independent of the event of getting a red ball on the second draw?

  1. Balls, Without Replacement

Five red balls and two balls are placed into a magical probability box. Two balls are selected without replacement.

  1. Without any prior knowledge, what is the probability the first ball is red?

  2. Without any prior knowledge, what is the probability the second ball is red?

  3. Without any prior knowledge, what is the probability the first ball is blue?

  4. Without any prior knowledge, what is the probability the second ball is blue?

  5. If the first ball is blue, what is the probability the second ball is red?

  6. If the first ball is red, what is the probability the second ball is red?

  7. If the first ball is blue, what is the probability the second ball is blue?

  8. If the first ball is red, what is the probability the second ball is blue?

  9. Based on your answers to part a through part h, is the event of getting a red ball on the first draw independent of the event of getting a blue ball on the second draw? Is the event of getting a blue ball on the first draw independent of the event of getting a red ball on the second draw?

  1. Coins

Solve these problems about flipping a fair coin.

  1. Suppose you flip a fair coin twice. What is the probability of getting two heads?

  2. Suppose you flip a fair coin three times. What is the probability of getting three heads?

  3. Suppose you flip a fair coin n times. What is the probability of getting n heads?

  4. Suppose you flip a fair coin three times. What is the probability of getting two heads?

  5. Suppose you flip a fair coin n times. What is the probability of getting two heads?

Hint

part d and part e require Combinatorics!

  1. Dice

Solve these problems about rolling a six-sided die.

  1. What is the probability of getting exactly three 6’s in three die rolls?

  2. What is the probability of getting atleast one 5 in four rolls?

Hint

For part b, first find the probability of getting no 5’s in four rolls and then use the Law of Complements to find the desired probability.

  1. A fair six-sided die is rolled twice. Let A denote the event that the sum of the outcomes is odd. Let B denote the event that it lands on 2 on the first toss. Are A and B independent? Why or why not?

  2. Ghahramani, 2005 Suppose that two numbers are selected at random and independently from the interval (0,1). What is the probability that the first one is less than \frac{3}{4}, and the second one is greater than \frac{1}{4}?

  3. Ghahramani, 2005 In a certain game, you perform three tasks. You flip a quarter, and win if you get heads. You roll a single die, and win if you get a six. You pick a card from a full playing-card deck, and win if you pick a card in the suit of spades. If any of these task are successful, then you win the game. What is the probability of winning?

Hint

You win in the case you get a head or you get a six or you get a spade. Don’t forget to account for the overlapping events!

  1. Ghahramani, 2005 In data communications, a message transmitted from one end is subject to various sources of distortion and may be received erroneously at the other end. A bit is the smallest unit of information transmitted, and is either 0 or 1. Suppose that a message of 64 bits is transmitted. If each bit is received incorrectly with a probability 0.0001 independently of the other bits, what is the probability the message is free of error?

A.P. Exam Practice#

  1. 2018 Free Response, #3

Approximately 3.5 percent of all children born in a certain region are from multiple births (that is, twins, triplets, etc.). Of the children born in the region who are from multiple births, 22 percent are left-handed. Of the children born in the region who are from single births, 11 percent are left-handed.

  1. What is the probability that a randomly selected child born in the region is left-handed?

  2. What is the probability that a randomly selected child born in the region is a child from a multiple birth, given that the child selected is left-handed?

  3. A random sample of 20 children born in the region will be selected. What is the probability that the sample will have at least 3 children who are left-handed?

  1. 2018, Free Response, #3

A medical researcher surveyed a large group of men and women about whether they take medicine as prescribed. The responses were categorized as never, sometimes, or always. The relative frequency of each category is shown in the table.

../../../_images/2019_apstats_frp_3.png

One person from those surveyed will be selected at random.

  1. What is the probability that the person selected will be someone whose response is never and who is a woman?

  2. What is the probability that the person selected will be someone whose response is never or who is a woman?

  3. What is the probability that the person selected will be someone whose response is never given that the person is a woman?

  4. For the people surveyed, are the events of being a person whose response is never and being a woman independent? Justify your answer.

  5. Assume that, in a large population, the probability that a person will always take medicine as prescribed is 0.54. If 5 people are selected at random from the population, what is the probability that at least 4 of the people selected will always take medicine as prescribed? Support your answer.

  1. 2017, Free Response, #3

A grocery store purchases melons from two distributors, J and K. Distributor J provides melons from organic farms. The distribution of the diameters of the melons from Distributor J is approximately normal with mean 133 millimeters (mm) and standard deviation 5 mm.

  1. For a melon selected at random from Distributor J, what is the probability that the melon will have a diameter greater than 137 mm?

  2. Distributor K provides melons from nonorganic farms. The probability is 0.8413 that a melon selected at random from Distributor K will have a diameter greater than 137 mm. For all the melons at the grocery store, 70 percent of the melons are provided by Distributor J and 30 percent are provided by Distributor K.

    1. For a melon selected at random from the grocery store, what is the probability that the melon will have a diameter greater than 137 mm?

    2. Given that a melon selected at random from the grocery store has a diameter greater than 137 mm, what is the probability that the melon will be from Distributor J?

  1. 2003, Free Response Form B, #2

A simple random sample of adults living in a suburb of a large city was selected. The age and annual income of each adult in the sample were recorded. The resulting data are summarized in the table below, where the rows represent the number in that age group and the columns represent the number in that income bracket,

$25,000 - $35,000

$35,001 - $50,000

Over $50,000

Total

21 - 30

8

15

27

50

31 - 45

22

32

35

89

46 - 60

12

14

27

53

Over 60

5

3

7

15

Total

47

64

96

207

  1. What is the probability that a person chosen at random from those in this sample will be in the 31-45 age category?

  2. What is the probability that a person chosen at random from those in this sample whose incomes are over $50,000 will be in the 31-45 age category? Show your work.

  3. Based on your answers to part a and part b, is annual income independent of age category for those in this sample? Explain.

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